Homework 13-solutions

# Homework 13-solutions - wtm369 Homework 13 Helleloid(58250...

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wtm369 – Homework 13 – Helleloid – (58250) 1 This print-out should have 25 questions. Multiple-choice questions may continue on the next column or page – fnd all choices beFore answering. 001 10.0 points Consider the Following Functions: F 1 ( x ) = x 2 + 2 x, - 1 x 1 , F 2 ( x ) = | x + 1 | , - 1 x 1 , F 3 ( x ) = sin 2 x, - π/ 4 x 4 . Which have an inverse on the given domain? 1. F 1 and F 2 only 2. F 1 only 3. none oF them 4. F 2 only 5. F 1 and F 3 only 6. F 2 and F 3 only 7. F 3 only 8. all oF them correct Explanation: A Function f has an inverse, f 1 , when any one oF the Following conditions are satisfed: (1) it is a 1-1 Function on its domain, (2) its graph passes the horizontal line test, (3) it is either strictly increasing or strictly decreasing. Consequently, A. Have: ( f strictly increasing on given do- main). B. Have: ( f ( x ) = x + 1 on domain, so f is strictly increasing). C. Have: (is 1-1). 002 10.0 points A Function f is known to have an inverse f 1 . Which oF the Following could be the graph oF y = f ( x )? 1. 2 4 - 2 - 4 2 4 - 2 - 4 2. 2 4 - 2 - 4 2 4 - 2 - 4 correct 3. 2 4 - 2 - 4 2 4 - 2 - 4 4. 2 4 - 2 - 4 2 4 - 2 - 4 5. 2 4 - 2 - 4 2 4 - 2 - 4

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wtm369 – Homework 13 – Helleloid – (58250) 2 Explanation: If f has an inverse it must be a one-to-one function; equivalently, its graph must pass the horizontal line test. The only graph having this property is 2 4 - 2 - 4 2 4 - 2 - 4 003 10.0 points Determine if f ( x ) = 9 - 8 x has an inverse, and Fnd this inverse if it does. 1. f 1 ( x ) = 9 8 + 1 8 x 2. f 1 ( x ) = 1 9 - 8 x 3. f 1 ( x ) does not exist 4. f 1 ( x ) = 1 8 x - 9 8 5. f 1 ( x ) = 9 8 - 1 8 x correct Explanation: A linear function with non-zero slope is either an increasing function or a decreasing function, depending on the sign of the slope. In either case, however, it is a 1-1 function, and so has an inverse. To determine this inverse when f ( x ) = 9 - 8 x we Frst solve for x in y = 9 - 8 x : x = 9 8 - 1 8 y , and then interchange x, y . Thus f 1 ( x ) = 9 8 - 1 8 x . 004 10.0 points ±ind the inverse of f ( x ) = 1 + 3 x 7 - 2 x . 1. f 1 ( x ) = 7 x - 1 3 x + 2 2. f 1 ( x ) = 7 x + 1 2 x + 3 3. f 1 ( x ) = 3 x - 1 2 x + 7 4. f 1 ( x ) = 7 x + 1 3 x + 2 5. f 1 ( x ) = 7 x - 1 2 x + 3 correct Explanation: To determine the inverse of f we Frst solve for x in y = 1 + 3 x 7 - 2 x . In this case x = 7 y - 1 2 y + 3 . The inverse f 1 is now obtained by inter- changing x, y . Thus y = f 1 ( x ) = 7 x - 1 2 x + 3 . 005 10.0 points ±ind the inverse function, g , of f when f ( x ) = 4 x 9 - x 2 , 0 x < 3 . 1. g ( x ) = 3 x 16 + x 2 , x 0
wtm369 – Homework 13 – Helleloid – (58250) 3 2. g ( x ) = 3 x 16 + x 2 , x 4 3. g ( x ) = 3 x 16 + x 2 , x ≤ - 4 4. g ( x ) = 3 x 16 + x 2 , x 0 correct 5. g ( x ) = 3 x 16 - x 2 , x 0 6. g ( x ) = 3 x 16 - x 2 , x 0 Explanation: Since f is 1-1 on its domain [0 , 3), its in- verse, g , exists. This inverse is found by interchanging x , y in f ( x ) = y = 4 x 9 - x 2 and solving for y . Thus we have to solve for y when x 2 = 16 y 2 9 - y 2 , i . e ., 9 x 2 - x 2 y 2 = 16 y 2 . Consequently, g ( x ) = 3 x 16 + x 2 , x 0 .

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Homework 13-solutions - wtm369 Homework 13 Helleloid(58250...

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